kallis@pen.DEC (06/06/85)
Steve Aldrich asks if there's a clearcut definition of a "dimension."
There are many definitions.
Primarily, a dimension is a measurement. Or something that can be
quantified. Normally, we think of dimensions in terms of mutually
perpendicular space "vectors" [in quotes to illustrate that I know
the difference between vectors and scalars] we usually term length
breadth/width and height (or as my old 3rd-grade teacher was wont
to say, "heighth"). Once we determine an arbitrary location as a
reference ("origin"), we can extend lines in the appropriate direc-
tions to establish these measures. These spatial dimensions we
measure in linear units such as inches, meters, furlongs, or light
years.
Time is a dimension, which we measure in seconds, minutes, hours, years,
and so on. Some models of the Universe make Time a "dimension" perpen-
dicular" to the space dimensions, but that's unnecessary for our discus-
sions.
Brightness or intensity is a dimension, usually measured in "magnitude."
At least in astronomical circles. It is also measured in foot-candles.
Temperature is a dimension, measured in degrees Kelvin, Celsuis, Farenheit,
or Rankine.
"Color" is a dimension, measured in Angstrom Units.
In short, anything that can be measured in definable units is a dimension.
With that all cleared away, most people talking about "dimensions" are
generally implying spatial ones. Discounting the time aspect, our uni-
verse appears to have three spatial dimensions that are curved, according
to some theories, in a fourth. There is no reason why there might not be
an infinite number of spatial dimensions, all at right angles to each other,
but there's no proof that any of them exist (or don't, for that matter).
Representing higher dimensions (if any) in three-dimensional space is as or
more difficult than representing three-dimensional space on a sheet of
paper (essentially two-dimensional).
I hope this has helped.
regards,
Steve Kallis, Jr.matt@oddjob.UUCP (Matt Crawford) (06/12/85)
In article <2529@decwrl.UUCP> kallis@pen.DEC (Steve Kallis, Jr.) writes: >Steve Aldrich asks if there's a clearcut definition of a "dimension." >There are many definitions. > >Primarily, a dimension is a measurement. Or something that can be >quantified. . . . >Time is a dimension, which we measure in seconds, minutes, hours, years, >and so on. . . . >Brightness or intensity is a dimension, usually measured in "magnitude." >At least in astronomical circles. It is also measured in foot-candles. >Temperature is a dimension, measured in degrees Kelvin, Celsuis, Farenheit, >or Rankine. Steve Kallis is (perhaps understandably) mixing the two usages of the word "dimension". One usage is synonymous with "units", as in: Velocity has dimensions of length/time Newton's constant has dimensions (length cubed)/(mass * time squared) The fine structure constant is dimensionless. The term "dimensionless units" is also employed to mean that all quantities have been multiplied by the appropriate powers of of the speed of light, Newton's constant, Boltzmann's constant and Planck's constant to cancel all the units. In effect, this sets all four of those constants equal to 1. The geometrical meaning of "dimension" does not depend on the existence of a set of mutually perpendicular coordinate axes. Loosely speaking, a manifold (such as space-time) has dimension n if each small region of the manifold can be put into a smooth 1-to-1 correspondence with a subset of the ordinary Euclidean space R^n. I hope this clarifies rather than the opposite. _____________________________________________________ Matt University crawford@anl-mcs.arpa Crawford of Chicago ihnp4!oddjob!matt
irenas@tekig4.UUCP (Irena Sifrar) (06/12/85)
Your dimensions are very intuitive and the ones with which we all deal every day. Yet there are other dimensions, for example Hausdorff dimension in mathematics. It is abstract, doesn't have much use for a typical man in the street, but helps mathematics. There are other, less intuitive dimensions, too, to which your definition does not apply. Irena Sifrar
apteryx@ucbvax.ARPA (Brian Peterson) (06/22/85)
In article <2529@decwrl.UUCP>, kallis@pen.DEC writes: > ... > generally implying spatial ones. Discounting the time aspect, our uni- > verse appears to have three spatial dimensions that are curved, according > to some theories, in a fourth. There is no reason why there might not be > an infinite number of spatial dimensions, all at right angles to each other, > but there's no proof that any of them exist (or don't, for that matter). > ... > Steve Kallis, Jr. I read in a recent Analog science article (~before June '85) that cosmologists are postulating the existence of about 7 more spatial dimensions to account for the actions of the 4 forces in an attempt to unify them all. The dimensions would be rather small, and "travelling" in them wouldn't make sense. Brian Peterson ...!ucbvax!brianp